# Introduction to Mathematical Physics/Statistical physics/Canonical distribution in classical mechanics

Consider a system for which only the energy is fixed. Probability for this system to be in a quantum state ${\displaystyle (l)}$ of energy ${\displaystyle E_{l}}$ is given (see previous section) by:

${\displaystyle P_{l}={\frac {1}{Z}}e^{-E_{l}/k_{B}T}}$

Consider a classical description of this same system. For instance, consider a system constituted by ${\displaystyle N}$ particles whose position and momentum are noted ${\displaystyle q_{i}}$ and ${\displaystyle p_{i}}$, described by the classical hamiltonian ${\displaystyle H(q_{i},p_{i})}$. A classical probability density ${\displaystyle w^{c}}$ is defined by:

eqdensiprobaclas

${\displaystyle w^{c}(q_{i},p_{i})={\frac {1}{A}}e^{-H(q_{i},p_{i})/k_{B}T}}$

Quantity ${\displaystyle w^{c}(q_{i},p_{i})dq_{i}dr_{i}}$ represents the probability for the system to be in the phase space volume between hyperplanes ${\displaystyle q_{i},p_{i}}$ and ${\displaystyle q_{i}+dq_{i},p_{i}+dp_{i}}$. Normalization coefficients ${\displaystyle Z}$ and ${\displaystyle A}$ are proportional.

${\displaystyle A=\int dq_{1}...dq_{n}\int dp_{1}...dp_{n}e^{-H(q_{i},p_{i})/k_{B}T}}$

One can show [ph:physt:Diu89] that

${\displaystyle Z={\frac {1}{(2\pi \hbar )^{3N}}}A}$

${\displaystyle 2\pi \hbar ^{N}}$ being a sort of quantum state volume.

Remark:

This quantum state volume corresponds to the minimal precision allowed in the phase space from the Heisenberg uncertainty principle:

\index{Heisenberg uncertainty principle}

${\displaystyle \Delta x\Delta p>\hbar }$

Partition function provided by a classical approach becomes thus:

${\displaystyle Z={\frac {1}{(2\pi \hbar )^{N}}}\int dq_{1}...dq_{n}\int dp_{1}...dp_{n}e^{-H(q_{i},p_{i})/k_{B}T}}$

But this passage technique from quantum description to classical description creates some compatibility problems. For instance, in quantum mechanics, there exist a postulate allowing to treat the case of a set of identical particles. Direct application of formula of equation eqdensiprobaclas leads to wrong results (Gibbs paradox). In a classical treatment of set of identical particles, a postulate has to be artificially added to the other statistical mechanics postulates:

Postulate:

Two states that does not differ by permutations are not considered as different.

This leads to the classical partition function for a system of ${\displaystyle N}$ identical particles:

${\displaystyle Z={\frac {1}{N!}}{\frac {1}{(2\pi \hbar )^{3N}}}\int \prod dp_{i}^{3}dq_{i}^{3}e^{-H(q_{i},p_{i})/k_{B}T}}$